How changing the cutoff value affects sensitivity, specificity, etc.
The cutoff value line is at the value where we’d call a test result lower than that “normal” and call a test result that was higher than that “abnormal”. That’s what we mean when we set a cutoff value. It’s the value that separates the test results we call “normal” from the test results we call “abnormal”.
Let’s consider a disease (e.g., diabetes) in which the lab values that are high (high blood glucose) indicate the presence of the disease and lab values which are low indicate the normal state (no disease).
Where are our TP, TN, FP, and FN subjects?
The white area under the “Diseased Population” graph (to the right of the cutoff value) are our True Positive (TP) numbers (diseased patients whose lab values are above the cutoff value).
The white area under the “Healthy Population” graph (to the left of the cutoff value are our True Negative (TN) numbers (healthy patients whose lab results are lower than the cutoff value)
The vertically hatched area to the left of the cutoff line is in the “Diseased Population” curve (patients actually have the disease, but is to the left of (lower than) the cutoff value, so their lab results would be called “normal” (because they are lower than the cutoff value). That area, then, represents our False Negative (FN) patients (have disease, but lab results are negative).
The lightly stippled area, under the “Healthy Population” curve and to the right of the cutoff value are our False Positive (FP) patients (healthy, but their lab values are above the cutoff value, so those lab values are positive, but the patients do not really have the disease.
So, if we move the cutoff value line to the right (toward the higher numbers), what happens to those four areas?
FP area (lightly stippled) gets smaller
FN area (vertical hatched) gets larger
TP area will decrease as the cutoff line cuts out more of the white area under the “Diseased Population” curve
TN area will increase because the FP portion of the “Healthy Population” curve is getting smaller
So for sensitivity:
sens = TP / (TP + FN), we’d have TP getting smaller and FN getting larger, so our numerator is decreasing and our denominator is staying the same (this is because (TP + FN) is the number of patients who actually have the disease and that hasn’t changed … all we’ve changed is the cutoff value of what test result we call “positive” and what test result we call “negative”. So, the decreasing numerator, with the same denominator means that our sensitivity will decrease.
Let’s look at that with a 2 x 2 table. We’ll start off with 50 patients who have the disease and 100 patients who are normal. At our first cutoff value, 35 patients who have the disease have positive test results that are called “positive” (TP). The remainder of the diseased patients (50 – 35) = 15 then had test results that were called “negative” and they will then be our False Negatives (FN).
Say that 5 patients who didn’t have the disease had lab results that were positive (FP), and the rest of the non-diseased patients (100 – 5) = 95 had test results that were negative (TN)
|Disease Yes = 50||Disease No = 100||Totals|
|Test result +||TP = 35||FP = 5||(TP + FP) = 40|
|Test result –||FN = 15||TN = 95||(TN + FN) = 110|
|Totals||(TP + FN) = 50||100||(TP + TN + FP + FN) = 150|
Our sensitivity of the test is:
TP / (TP + FN) = 35 / 50 = 0.7, or 70%
We move our cutoff value upwards, which decreases our TP number (see above where I describe that change with reference to Fig 3.2 on page 76 of the text). Say the TP goes down to 30. For calculating sensitivity, we don’t really need the TN and FP numbers, but let’s say they change with FP decreasing from 5 to 2, then we have the following 2 x 2 table:
|Disease Yes||Disease No = 100||Totals|
|Test result +||TP = 30||FP = 2||(TP + FP) = 32|
|Test result –||FN = 20||TN = 98||(TN + FN) = 118|
|Totals||(TP + FN) = 50||(TN + FP) = 98||(TP + TN + FP + FN) = 150|
Now our sensitivity = TP / (TP + FN) = 30 / 50 = 0.6 or 60%
The sensitivity has diminished.
You can go through analogous cases to see the change in specificity; you can also do the calculations with the cutoff value moved to the left (toward the lower numbers), make the appropriate changes in your TP, TN, FP, and FN values and see how sensitivity and specificity would be altered. Just remember that you can’t change the total number of diseased and non-diseased patients. Those numbers remain constant and are the totals in columns one (TP + FN) and two ((TN + FP).
Hope that helps!